Solving the ODE y' = x^2 + y^2

  • Thread starter Benny
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In summary, the equation y = \frac{dy}{dx} can be solved by substitution using y = \frac{u^{'}}{-u} and turning the crank.
  • #1
Benny
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Hi I'm just wondering if there is a way to solve the ODE: y' = x^2 + y^2. I've skimmed through my book and I haven't found a way to do this. Any help appreciated.
 
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  • #2
i suppose that y' = [tex]\frac{dy}{dx}[/tex] ?


Is x function of y ? is y function of x ? Or both x and y function of some other argument ?

Your are going to need to be a bit more specific here


marlon
 
  • #3
I think if we are provided with initial conditions Picard's iteration can come up with the required answer.
 
  • #4
I found this question while I was going through some exam papers yesterday. I'm pretty sure that in this question y is a function of x, or at least that's what the question seems to imply. I was just curious as to whether there is a standard way to solve a question like that.
 
  • #5
Benny said:
Hi I'm just wondering if there is a way to solve the ODE: y' = x^2 + y^2. I've skimmed through my book and I haven't found a way to do this. Any help appreciated.

Hey Benny, you got this? It's a particular form of the Riccati Equation right?

[tex]y^{'}+Q(x)y+R(x)y^2=P(x)[/tex]

so yours is:

[tex]y^{'}-y^2=x^2[/tex]

Thus make the substitution:

[tex]y=\frac{u^{'}}{-u}[/tex]

Turn the crank and get:

[tex]u^{''}-x^2u=0[/tex]

Then solve via power series, then take the derivative, form the quotient, then a plot. Put it all into Mathematica and back-substitute to make sure it's correct, then compare with numerical results. Or just do what you want. :smile:
 
  • #6
Thanks for your response Saltydog. I haven't seen this type of equation before and I've done only very basic DE questions with series, most of which I've forgotten by now, so it'll be a while(ie. during my 2 week break which starts next week) until I have a look at series solutions again and try to get this one out.
 

1. What exactly is an ODE?

An ODE, or ordinary differential equation, is a mathematical equation that involves an unknown function and its derivatives. It is used to model various physical phenomena and is an important tool in many scientific fields.

2. How do you solve an ODE?

Solving an ODE involves finding the function that satisfies the given equation. This can be done analytically using mathematical techniques such as separation of variables, substitution, or integration. It can also be solved numerically using computer algorithms.

3. What is the purpose of solving the ODE y' = x^2 + y^2?

The purpose of solving this ODE is to find the function y(x) that satisfies the equation. This function can then be used to model a physical system or phenomenon that exhibits behavior described by this differential equation.

4. What are some real-life applications of solving ODEs?

ODEs have many real-life applications in fields such as physics, engineering, economics, and biology. They can be used to model population growth, chemical reactions, electrical circuits, fluid dynamics, and more.

5. Are there any challenges in solving ODEs?

Yes, there can be challenges in solving ODEs, especially when dealing with complex or nonlinear equations. It may be difficult to find an analytical solution, and numerical methods may be required. Additionally, the initial conditions and parameters of the ODE must be accurately determined for the solution to be meaningful.

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